Can I create a noise texture with a "darker end"?
You can add multiple materials to an object in Blender. In Edit Mode just select the faces of the outer part and assign them to a new material. In the example I've done this for the outer, middle (white line), and the inner part.
The &quot;darker end&quot; can be created with a ColorRamp and a MixRGB node in the node setup. Set the mode of the MixRGB node to Multiply and factor to 1.0. To move the &quot;darker end&quot; of the outer part to the bottom right I had to add a Mapping node and rotate the position a bit.

The shading for the inner area looks similar. Just swapped the grey values of the  ColorRamp to darken the top right part of this area a bit.

If the colors appear too dark you might need more light in your scene. The logo on the screenshot is 3m tall (to get the shading right) and there is a 3000W point light, 3m radius (affects the shadow softness).
With a Solidify modifier you can add some depth/3D to your logo and it gets some nice light reflections. Add a plane as background to see the drop shadow if you want.
I removed the Alpha channel since it's always 1 and it had no use(?). If you activate the Node Wranger addon that ships with Blender you can press CTRL + SHIFT + LMB to get a preview of the selected node. This can help a lot to see what's going on.